Is it reasonable to first extract the lower-level factor scores from a 2-level CFA (unstructured higher-level covariance matrix, acceptable model fit), and then assign a mixed model to the factor (so the factor is treated as dependent variable and assumed to have a multilevel structure)?

An example (not real) is: measuring IQ of students coming from different schools using some questionnaire. Since the interest is in the student-level common factor (IQ), we run a 2-level CFA with school-level having an unstructured covariance matrix. If the model fits well, then we extract the factor score for each student and assume it is the IQ estimate. Next, we want to model IQ with some observed covariates to see if IQ is affected by them. Since again the students come from different schools, we run a 2-level regression model.

what if the ICC (for the extracted factor scores) is above 0.1 and we are interested in parameters in the structural part, is it better to take into account the 2-level structure in order to obtain correct standard error estimates?

Maybe I did not make this very clear. Let's say the true IQ for each student "i" in school "j" is T_{ij}, and if there exists a multilevel structure in the real case, T_{ij} could be rewritten as T_{ij}=T_{within}+T_{between}.

Does the student-level factor score (defined in the first message) reflect the true IQ, i.e. T_{ij}, or the within part of it, i.e. T_{within}? If it reflects the latter, how can we get an estimate of the true IQ T_{ij} ?

In some cases, the factor structure may not be the same for each level, e.g. 2 factors at lower level and only 1 factor at higher level, or each level has 2 factors but are loaded on by different items across levels. I think in this case only the T_{within} could be estimated for each of the lower-level factors. The T_{between} of these 2 lower-level factors are mixed to some extent and may not be possible to split them up.